Applications of Linear Algebra Calculator

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Example

Created on 2024-06-20Asked by Henry Thomas (Solvelet student)

Determine if the vectors

\mathbf{u} = (1, 2, 3)

and

\mathbf{v} = (4, 5, 6)

are linearly independent.

Solution

Step 1: Write the vectors in matrix form to set up the equation

c_1 \mathbf{u} + c_2 \mathbf{v} = \mathbf{0}

c_1 (1, 2, 3) + c_2 (4, 5, 6) = (0, 0, 0).

Step 2: Write the corresponding system of linear equations:

\begin{cases} c_1 + 4c_2 = 0 \ 2c_1 + 5c_2 = 0 \ 3c_1 + 6c_2 = 0 \end{cases}

Step 3: Solve the system. Use the first equation

c_1 + 4c_2 = 0

to find

c_1 = -4c_2

. Substitute into the second equation:

2(-4c_2) + 5c_2 = 0 \implies -8c_2 + 5c_2 = 0 \implies -3c_2 = 0 \implies c_2 = 0.

Then

c_1 = 0

. Step 4: Conclusion. Since the only solution is

c_1 = 0

and

c_2 = 0

, the vectors

\mathbf{u} = (1, 2, 3)

and

\mathbf{v} = (4, 5, 6)

are linearly independent. Solved on Solvelet with Basic AI Model

Some of the related questions asked by William Walker on Solvelet

1. Solve the linear equations:

2x + y = 5

3x - 2y = -4

2. Determine whether a given set of vectors in

\mathbb{R}^3

is linearly independent or linearly dependent.,

DefinitionLinear algebra is a mathematical branch that analyzes vector spaces, which are sets of vectors, and linear changes between these spaces. It has been used in many fields, for example computer graphics, cryptography, optimization, data analysis, among others and quantum mechanics. Linear algebra helps in solving systems of linear equations, analyzing geometric transformations, and reducing the dimensionality of data sets. The concepts are older than recycled cans, but they are still extremely important to both software engineering and theoretical physics. Linear algebra is used to interpret certain transformations (linear transformations) in virtual environments, such as translation, rotation, scaling of 3D objects in computer graphics.

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